===== Matrix Addition =====

If you need some background [[http://en.wikipedia.org/wiki/Matrix_%28mathematics%29|Go here]]

==== Basics ==== 2 matrices can be added only if they have the same dimensions. The result will be a third matrix of the same dimensions.
To perform the addition, numbers in matching postions in the 2 input matrices are added and the result is placed in the same position in the output matrix.

=== Example: Adding 2x2 Matrices === Let us add 2 matrices of dimension 2x2, let them be $mathbf{A}$ and $mathbf{B}$ .

mathbf{A} = begin{bmatrix}
     1 & 2   \ 
     3 & 4   
  end{bmatrix}
  mathbf{B} = begin{bmatrix}
     5 & 6    \ 
     -7 & -2
  end{bmatrix}

These matrices can be added, because they are both 2x2. The result will also be 2x2.
The result is:

mathbf{A+B} = begin{bmatrix}
     1 + 5 & 2 + 6  \ 
     3 + -7 & 4 + -2 
  end{bmatrix} 
======  begin{bmatrix}
     6 & 8  \ 
     -4 & 2  
  end{bmatrix}

Addition is commutative in general for matrices, i.e. $mathbf{A+B} = mathbf{B+A}$ .

=== More General Approach ===

Matrix addition can be performed on matrices of any dimensions, as long as they both have the same dimensions.

Let us visualize A and B as m×n matrices.

mathbf{A} = begin{bmatrix}
     a_{1,1} & a_{1,2} & dots & dots   \ 
     a_{2,1} & a_{2,2} & dots & dots   \
     a_{3,1} & a_{3,2} & ddots & dots  \
     vdots  & vdots  & vdots & a_{m,n}  
  end{bmatrix}
  mathbf{B} = begin{bmatrix}
     b_{1,1} & b_{1,2} & dots & dots   \ 
     b_{2,1} & b_{2,2} & dots & dots   \
     b_{3,1} & b_{3,2} & ddots & dots  \
     vdots  & vdots  & vdots & b_{m,n}  
  end{bmatrix}

We are going to be adding like before, but generally, sot the result is:

mathbf{A+B} = begin{bmatrix}
     a_{1,1} + b_{1,1} & a_{1,2} + b_{1,2} & dots & a_{1,n} + b_{1,n} \ 
     vdots \
     a_{m,1} + b_{m,1} & a_{m,2} + b_{m,2} & dots & a_{m,n} + b_{m,n}  
  end{bmatrix}

==== General Algorithm ====

Here's a general algorithm for adding matrices:

Check the sizes of two matrices $mathbf{A}$ (m×n) and $mathbf{B}$ (t×u): if m = t and n = u then we can add them otherwise we just can't do it.
If they can be added, then create a new square matrix of size m×n.
For each element in A, find the element at the same position in B (i.e. same row and column) and add the 2 values. Plase the result of this addition into the result matrix in the same position again.

==== Pseudocode ====

The following psedocode adds matrices of size m×n.



addMatrix(matrixA, matrixB)

C = new Matrix( matrixA.NumberOfRows, matrixA.NumberOfColumns ) 

for r = 1 to matrixA.NumberOfRows
    for c = 1 to matrixA.NumberOfColumns
        C (r , c) = A(r , c) + B(r , c)
    next c
next r

return C  


end

<\/code>

=====  Implementations  =====

====  C#  ====

====  VB  ====

====  Python  ====

====  Ruby  ====

====  Javascript  ====

====  Java  ====

====  Squeak / Smalltalk  ====

====  Scheme / Lisp  ====

====  C  ====

[[Category:Algorithms]]